Abstract: The interaction of a periodically inhomogeneous electron beam of finite transverse extent with its surrounding plasma is investigated. The hydrodynamic model is adopted for both the beam, which is of finite transverse extent, and the plasma, which can assume either an infinite or a finite transverse extent. The results indicate that for zero external magnetic field no stabilizing effects occur. This is in contrast to the results of the one‐dimensional analysis previously reported. The dispersion relation for the finite magnetic field case is also presented and discussed.